Let the probability of a family owning a dishwasher be \(p = 0.22\). The number of families is \(n = 15\).

We need to find \(P(4 \, \text{to} \, 6)\), which is the probability that between 4 and 6 families own a dishwasher.

Using the binomial probability formula:

\(P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}\)

Calculate for \(k = 4, 5, 6\):

\(P(4) = \binom{15}{4} (0.22)^4 (0.78)^{11}\)

\(P(5) = \binom{15}{5} (0.22)^5 (0.78)^{10}\)

\(P(6) = \binom{15}{6} (0.22)^6 (0.78)^9\)

Sum these probabilities:

\(P(4, 5, 6) = \binom{15}{4} (0.22)^4 (0.78)^{11} + \binom{15}{5} (0.22)^5 (0.78)^{10} + \binom{15}{6} (0.22)^6 (0.78)^9\)

\(= 0.398\)